601 research outputs found
Building spanning trees quickly in Maker-Breaker games
For a tree T on n vertices, we study the Maker-Breaker game, played on the
edge set of the complete graph on n vertices, which Maker wins as soon as the
graph she builds contains a copy of T. We prove that if T has bounded maximum
degree, then Maker can win this game within n+1 moves. Moreover, we prove that
Maker can build almost every tree on n vertices in n-1 moves and provide
non-trivial examples of families of trees which Maker cannot build in n-1
moves
Generating random graphs in biased Maker-Breaker games
We present a general approach connecting biased Maker-Breaker games and
problems about local resilience in random graphs. We utilize this approach to
prove new results and also to derive some known results about biased
Maker-Breaker games. In particular, we show that for
, Maker can build a pancyclic graph (that is, a graph
that contains cycles of every possible length) while playing a game on
. As another application, we show that for , playing a game on , Maker can build a graph which
contains copies of all spanning trees having maximum degree with
a bare path of linear length (a bare path in a tree is a path with all
interior vertices of degree exactly two in )
Positional Games
Positional games are a branch of combinatorics, researching a variety of
two-player games, ranging from popular recreational games such as Tic-Tac-Toe
and Hex, to purely abstract games played on graphs and hypergraphs. It is
closely connected to many other combinatorial disciplines such as Ramsey
theory, extremal graph and set theory, probabilistic combinatorics, and to
computer science. We survey the basic notions of the field, its approaches and
tools, as well as numerous recent advances, standing open problems and
promising research directions.Comment: Submitted to Proceedings of the ICM 201
Efficient winning strategies in random-turn Maker-Breaker games
We consider random-turn positional games, introduced by Peres, Schramm,
Sheffield and Wilson in 2007. A -random-turn positional game is a two-player
game, played the same as an ordinary positional game, except that instead of
alternating turns, a coin is being tossed before each turn to decide the
identity of the next player to move (the probability of Player I to move is
). We analyze the random-turn version of several classical Maker-Breaker
games such as the game Box (introduced by Chv\'atal and Erd\H os in 1987), the
Hamilton cycle game and the -vertex-connectivity game (both played on the
edge set of ). For each of these games we provide each of the players with
a (randomized) efficient strategy which typically ensures his win in the
asymptotic order of the minimum value of for which he typically wins the
game, assuming optimal strategies of both players.Comment: 20 page
Fast strategies in biased Maker--Breaker games
We study the biased Maker--Breaker positional games, played on the
edge set of the complete graph on vertices, . Given Breaker's bias
, possibly depending on , we determine the bounds for the minimal number
of moves, depending on , in which Maker can win in each of the two standard
graph games, the Perfect Matching game and the Hamilton Cycle game
Tree universality in positional games
In this paper we consider positional games where the winning sets are tree
universal graphs. Specifically, we show that in the unbiased Maker-Breaker game
on the complete graph , Maker has a strategy to occupy a graph which
contains copies of all spanning trees with maximum degree at most ,
for a suitable constant and being large enough. We also prove an
analogous result for Waiter-Client games. Both of our results show that the
building player can play at least as good as suggested by the random graph
intuition. Moreover, they improve on a special case of earlier results by
Johannsen, Krivelevich, and Samotij as well as Han and Yang for Maker-Breaker
games
Fast Strategies in Waiter-Client Games on
Waiter-Client games are played on some hypergraph , where
denotes the family of winning sets. For some bias , during
each round of such a game Waiter offers to Client elements of , of
which Client claims one for himself while the rest go to Waiter. Proceeding
like this Waiter wins the game if she forces Client to claim all the elements
of any winning set from . In this paper we study fast strategies
for several Waiter-Client games played on the edge set of the complete graph,
i.e. , in which the winning sets are perfect matchings, Hamilton
cycles, pancyclic graphs, fixed spanning trees or factors of a given graph.Comment: 38 page
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